Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 155

as rapidly as we canwrite. Thus scientists differentiate polynomials with

gusto. Using (3.563) with n = -j gives

d 1 __ d x_/=

dx / dx

when x>0, and using it with n = .gives

d xVx= dx34 = 3x34

dx dx 2

when x> 0.

The last formula (3.566) can be remembered for years with the aid of a

little trick. We remember that the derivative of a quotient is a bigger

and better one and begin by drawing a long line to separate the numerator

from the denominator. We continue by putting v2 in the denominator

and then, while the v is in mind, begin the numerator by writing v. This

starts things right, and the rest can be remembered.

In our proof of the theorem, we fix (or select) an x in the domain of the

functions and put u = u(x), v = v(x), u + Au = u(x + Ax),

so that

V + Av = v(x + Ax)

du_ Auu= u (x + Ax) - u (x)

TX a +o Ox loo Ax

dv= lim Av

= lim

v(x + Ax) - v(x)

TX s. ..o Ax 10 Ax

We prove (3.561) and (3.562) together by starting with

y = Cu + civ,

where c and ci are constants; we can put c = cl = 1 to get (3.561) and we

can take ci = 0 to get (3.562). Then

y + AY = c(u + Du) + ci(v +Av)

and subtraction gives

Ay = C Du + ci Av.

Hence

Ox

= C

Ox+ ci Ax

The hypothesis of Theorem 3.56 implies

lim

Au

Ax=

du

l.o Ax =ax'